Page - 65 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Figure6.BrokensymmetryongeometricheatQduetoadjointactionof thegroupontemperatureβas anelementof theLiealgebra. ForHamiltonian,actionsofaLiegrouponaconnectedsymplecticmanifold, theequivarianceof themomentmapwithrespect toanaffineactionof thegrouponthedualof itsLiealgebrahasbeen studiedbyMarleandLibermann[100]andLichnerowics [101,102]: Theorem 2 (Marle Theorem on Cocycles). Let G be a connected and simply connected Lie group, R :G→GL(E) bea linear representationofG inafinite-dimensionalvector spaceE,and r : g→ gl(E) be the associated linear representation of its Lie algebra g. For any one-cocycleΘ : g→E of the Lie algebra g for the linear representation r, there exists aunique one-cocycle θ :G→E of theLie groupG for the linear representationRsuch thatΘ(X) = Teθ(X(e)),whichhasΘ as associatedLie algebra one-cocycle. TheLie groupone-cocycleθ is aLiegroupone-coboundary if andonly if theLiealgebraone-cocycleΘ is aLiealgebra one-coboundary. LetGbeaLiegroupwhoseLiealgebra isg. Theskew-symmetricbilinear form Θ˜ong=TeG can beextendedintoacloseddifferential two-formonG, since the identityon Θ˜meansthat itsexterior differentialdΘ˜vanishes. Inotherwords, Θ˜ isa2-cocycle for therestrictionof thedeRhamcohomology ofG to left invariantdifferential forms. In theframeworkofLiegroupactiononasymplecticmanifold, equivarianceofmomentcouldbestudiedtoprovethat there isauniqueactiona(.,.)of theLiegroupG onthedualg∗of itsLiealgebraforwhichthemomentmap J isequivariant, thatmeansforeachx∈M: J ( Φg(x) ) = a(g, J(x))=Ad∗g(J(x))+θ(g) (41) whereΦ :G×M→M is anactionofLiegroupGondifferentiablemanifoldM, the fundamental fieldassociatedtoanelementXofLiealgebragofgroupG is thevectorsfieldXMonM: XM(x)= d dt Φexp(−tX) (x) ∣∣∣∣ t=0 (42) withΦg1 ( Φg2(x) ) =Φg1g2(x)andΦe(x)= x.Φ isHamiltonianonasymplecticmanifoldM, ifΦ is symplecticandif forallX∈ g, the fundamentalfieldXM isgloballyHamiltonian. Thecohomology classof thesymplecticcocycleθonlydependsontheHamiltonianactionΦ, andnoton J. InAppendixB,weobserve thatSouriauLiegroupthermodynamics iscompatiblewithBalian gauge theoryof thermodynamics [103], that isobtainedbysymplectization indimension2n+2of contactmanifold indimension2n+1.Allelementsof theSouriaugeometric temperaturevectorare multipliedbythesamegaugeparameter. WeconcludethissectionbythisBourbakistecitationof Jean-MarieSouriau[34]: 65